The problem asks (teachers) to suggest an activity of three questions to students based on this picture to show that $\lim\limits_{x \to +\infty} \dfrac{x}{e^x}$
(picture : part of the hyperbola associated to $x \longmapsto \frac 1x$)
The picture suggests working with the following definition of the natural logarithm : $\ln(a) := \displaystyle\int_1^a \dfrac 1t dt$.
I considered showing first that $\lim\limits_{t \to +\infty} \frac{\ln t}{t} = 0$ then deduce the desired limit by a change of variable.
The problem is the following :
I first considered upbounding the area below the dark-shaded area by the area of the right shaded triangle but this gives the loose inequality $0<\ln(a)<\frac a2$ which isn't quite useful to conclude.
I thought about concavity using the equation of the chord joining points $(1,1)$ and $\left(a,\frac 1a \right)$ then integrated the concavity inequality to get : $\ln(a) \le \frac 12\left( a - \frac 1a\right)$ which doesn't give a proper squeezing argument to conclude for $a>1$ since we only get :
$$0<\frac{ln(a)}{a} < \frac 12 \left( 1 - \frac{1}{a^2}\right)$$
Thanks in advance for any clues, suggestions.